Last time I demonstrated that if a weeble of uniform density existed, then it could not be effectively 2D. This then led to considerations of the stability of 3D shapes in order to try and find the elusive shape with exactly one stable equilibrium point and one unstable equilibrium point. Since we can construct shapes with lots of stable points what we would really like to do is find a find of reducing this number. However, today, we take a sleight detour and consider the opposite problem of making an unstable point stable!
We start with a challenge (supposedly) set by Christopher Columbus. Whilst dining one night, a nobleman approached him and suggested that finding the Americas was not that impressive because anyone sailing out that way could not have missed them. In reply Columbus challenged the nobleman to take a normal egg and place it so it stood upright. Now, there are many ways of doing this, but most use some other piece of apparatus (such as alternating current), but Columbus used only the egg.
When the nobleman gave up Columbus simply took the egg and tapped it gently breaking the top of the shell, making it flatter. The moral of the story is normally anyone can solve a problem once they have seen a solution. However, for us, the story shows how to make unstable points stable.
Figure 2. Stabilising the unstable. By flattening the top the egg becomes stable upside down.
What is more amazing is that you can actually rigorously construct a mathematical algorithm that does exactly this flattening operation. Thus, if a uniform density weeble does exist then following this algorithm will allow us to construct a solid with any number of stable equilbria.
In two weeks we return to the trail of finding this illusive shape.
Last week I introduced the idea of stable and unstable equilibrium points using weebles as an example. As I mentioned, they work because their bottom is much heavier than the rest of the toy. However, suppose now that the weeble is constructed from a completely uniform material. Thus, the bottom would only be heavier if it had a larger volume of substance was put there. Would the weeble still work? This is the question we will be starting to look at this week.
It may surprise you but the simple answer is actually in your fridge! An uncooked egg’s density is pretty constant throughout its shape. There are small differences between the shell, albumen and yolk but they are unimportant and it is obvious to see that eggs don’t stand to attention [1]. The eggs actually have an infinite number of equilibrium points because they can be rolled over to any point on their side and they won’t move.
So a uniform egg shape won’t stand up, but will any shape? Mathematically, what we are looking for is a shape that has exactly one stable equilibrium point and one unstable equilibrium point.
Before we consider the 3D case, let us remove a dimension and consider flat 2D shapes. For two-dimensional convex shapes it can be proven that:
all planar, convex, homogenous shapes have at least 2 stable and 2 unstable equilibria.
A sketch of the proof can be found below. Hence, we know that if a weeble of uniform density exists then it cannot be effectively two-dimensional. What we can observe is that the 2D oval has two stable equilibria and for any number greater or equal to three the regular polygon with “n” sides will have “n” stable equilibria.
But what about three dimensions? Sadly, things are not so simple. However, when logical proofs are not forthcoming mathematicians are just as open to experimentation as any other scientist.
Below are a number 3D shapes, have a go at counting the number of stable equilibria they have. The answers can be found if you scroll down past the theorem proof.
Figure 2. A cube, a square based pyramid and a cylinder with edges sliced at an angle.
Now that you have got a feel for finding stable equilibria, have a think about what kind of shape would have one stable equilibrium point and one unstable only. Are you sure it even exists? Before we answer this question we take a detour next week and show how we can make our eggs not so lazy!
Consider a 2D convex object that has only one stable and one unstable equilibrium point. Define a function, R(ɵ), that is the distance of the perimeter away from the centre of gravity as shown in the figure on the left. The function depends on the angle ɵ (measured in radians) around the centre of gravity and so as ɵ increases from 0 to 2π the function, R, will do one revolution of the shape.
Now consider the graph of R. By assumption there are only two equilibria, one stable and one unstable, thus, the corresponding graph has just one maximum and one minimum, as shown in the figure on the right. This is how the equilibria where defined last week.
Suppose we now drop a horizontal line across the graph. In particular let the points at which the horizontal line touches the graph be separated by an angular distance of π radians. This will correspond to a straight cut through the shape. Call this corresponding value R0.
By definition everything above this horizontal would be the part of the shape that is further than R0 away from the centre of gravity and everything below would the part of the shape closer than R0 to the centre of gravity. This means that one half of the shape would be bigger than the other half, and, thus, heavier. But by definition the cut goes through the centre of gravity and so each side weighs the same. As we come to a contradiction, our original hypothesis (that such a shape exists) must be wrong.
The cube has 6 stable equilibria; one on each of the flat faces, as shown.
The cube has 5 stable equilibria; one on each of the flat faces, as shown.
The sliced cylinder only has 1 stable equilibrium. However, it is not a shape that satisfies our need because it has 3 unstable equilibria; one on the top of the cylinder and one at each of the tips.
Did you ever have set of weebles? I did. They were wonderful little toys that lived up to their advertising slogan of “weebles wobble, but they don’t fall down”. The whole premise was that you got a set of several small plastic egg shaped characters (as seen in Figure 1) that would always wobble back to an upright position, no matter what position you started them from. As you can imagine hours of fun could be had with these little things.
The toy's mechanism is very simple. The bottom of the weeble is much heavier than the rest of the body, this means its centre of gravity is very low. Due its egg shaped body the weeble will always wobble such that its centre of gravity is at its lowest point. The applet below shows how this idea works [1]. On the left shape click on the red star which is the centre of mass and move it around the shape. The effect can be seen in the shape on the right. The shape on the right can then be grabbed and started from different initial points. However, it will always evolve to a stable point.
Why am I talking about weebles? Well, they are a really nice way of demonstrating how mathematicians understand equilibrium points, stability and instability. Although theoretically if a body was placed at its equilibrium points nothing further would occur, we know that physically we cannot be that accurate. Thus, equilibrium points can then be separated (generally) into stable and unstable points [2]. Thankfully, the mathematical notion of stability accords with our everyday use of the word. A body is stable if given a small push (or perturbation) it returns back to its previous position.
Figure 2. A wobbling weeble.
For a concrete example of this consider Figure 3, which is a schematic diagram of a curved bowl containing some balls. A is a stable equilibrium point; if the ball is given a small push it will return back to A. If the ball is given a big enough push it can roll right over the dividing line into the C position and this is also a stable equilibrium point. Now consider placing a ball at B. Since the bowl is completely flat there B is also an equilibrium point. However, it is an unstable equilibrium point. This is because no matter how small a push the ball is given it will never head back to B, but rather roll to one of A or C. Importantly, stability is only defined at points A, B and C because these are the only equilibrium points.
Figure 3. Illustrating stable and unstable equilibrium points.
A similar idea is applicable to the states of the weeble. When the weeble is upright it is equivalent to the ball being at A or C; the weeble is at a stable equilibrium point. Now suppose you could balance the weeble perfectly on its head, such that the centre of gravity, G, is directly above the flat point of its head, as in Figure 4. This is equivalent to the point B in Figure 2. It is an equilibrium point and, so, theoretically, if placed there, with no perturbation, then the weeble would stay there forever. However, any perturbation, no matter how small, will cause it to flip over and turn the right way up.
Figure 4. Stable and unstable weebles.
In two weeks time I will continue this topic and see what happens when we remove the possibility of weighting the bottom and only depend on the geometry of the object.
[2] Note that there are also saddle points and centres but we’re not going to talk about them as they’re special cases that few people care about. Also, the stable and unstable points could each be further split up into oscillatory and non-oscillatory points but again we’re going to ignore this complexity.
Who wouldn’t want to be Santa? You work one day a year, people give you mince pies and you spend the rest of your time deciding who’s naughty and who’s nice. However, Santa’s job is more difficult than you might think and he is certainly a mathematical genius who could win $1,000,000!
Efficient packing is an Elf's forte.
Let’s focus on just two of his important tasks. Firstly, he has to pack his sleigh full of toys. Annoyingly, toys are not all the same shape so how does he efficiently pack his sleigh? Either he is excellent at Tetris, or he has managed to solve an extremely difficult maths problem known as “bin packing”. The problem is, simply stated, ‘given a fixed amount of space and packages of different sizes, is there a complete list of instructions that will allow you to make the most of the space?’
Santa's problems are bigger than his belly.
The second problem he faces is finding the quickest route around Earth that visits every house. This is known as the “travelling salesperson problem”. Is it possible to find a recipe that will always give you the fastest route?
In both cases no one knows the answer. What is worse is that we don’t even know if there is an answer! Mathematicians describe these problems as “NP” (Not Polynomial). This means that the difficulty of finding the best solution increases dramatically whenever you add another package or house.
Although these problems may be placed in fantasy, real life logistic companies face these obstacles every day and solving them would save a fortune. In fact, the solutions are so important that the Clay Mathematics Institute offers one million dollars to anyone who can either produce a method that works flawlessly, or show that one doesn’t exist.
So until Santa decides to retire and give up his secrets, that million is waiting for the right mind. Who knows? It could even be yours.
Merry Christmas from the Laughing Mathematician and see in 2013 for a new set of posts on the gömböc.
A very seasonal Turing pattern.
Addendum
Although we may not able to solve the problem yet we can still take a peak at Santa's solution.The United States and Canada aerospace defence organisation, known as NORAD, take it upon themselves to track Santa every year as he travels around the globe. You can follow Santa's journey through their website, http://www.noradsanta.org/.
I've recently been involved with a new mathematical project called Everything is Mathematical. A group of Spanish Mathematicians have written a comprehensive set of popular maths books, which have proved so successful that they're being translated into other languages, including English. The team behind the series have joined up with The Times newspaper and Marcus du Sautoy to present the series to the British audience.
But where do I fit it? Well, to generate some interest in the series they're producing a number of maths challenges that can be found on their website (mine is below).
Some of the challenges are
fairly standard and well known, however, there are a couple of gems
in there that I had never seen before! Keep a look out on the
website for the weekly challenges. If you do solve one of the problems there is a
competition you can enter. The winners receive a free subscription to the
entire set of books.
Now since there are 40 in the set and each one is £10, you may need come convincing before you part with your cash. To get you interested I thought I would review one of the books. Of course since I'm working with these guys you'll have to weight my review with the fact that I'm biased :) (although I'll try to remain neutral).
Who are they for?
Based on what I've read I think these books are more suitable for an
older audience. It could certainly appeal to those who were
educated in numerical subjects and then left due to career decisions, but never lost their interest, as
well as those who never really got on with mathematics but realise its
importance. Having said that I could also also see parents buying a set to enhance their children's education as school as well as benefit their own interests.
Appearance
The first thing I noticed about the series is the sheer comprehensiveness of the books. To be honest, I'd be hard pressed to even name 40 different mathematical fields! The books cover all the standard aspects such as geometry, symmetry, probability, etc. but then goes further with books on the mathematics of our senses and perception, artificial intelligence and, of course (my favourite) mathematical biology.
You may think (as I did) that, by spanning so many subjects, each one would be a pamphlet in size. Here, again, I am pleased to say I was surprised. Each book I've seen is a really nice hard back spanning over 100 pages in length. They contain: nice clear diagrams; formulas; anecdotes about the history; fact boxes and even problems to challenge you.
Book number 1.
What about the content?
Well here is my first admission: I haven't actually read the first book. The first one is all about the golden ratio and its geometry and to be honest, I've never been a fan of the golden ratio because even though it has some extremely nice mathematical properties I find that discussions quickly descend into numerology [Rant 1]. So, instead of starting with The Golden Ratio, I jumped straight in with "When Straight Lines Meet" a book all about non-Euclidean geometry.
Now I would've expected the book to have started with easy stuff, y'know setting up the cartesian coordinates and linear geometry leading on to hyperbolic and elliptic geometry later. How wrong could I have been? The first chapter drops you straight in by explaining the taxi cab metric, using the city of el Ensanche as an illustration.
el Ensanche's very regular city set up.
This is certainly one of the strengths of the books. They use of definite, tangible examples allowing them to discuss complex ideas and very quickly outstrip even my knowledge! For example I had never heard of Girolamo Saccheri's contribution to non-Euclidean geometry.
Each chapter is completely different and could potentially be read independently. For getting a deep overview about a specific aspect (history of geometry, curved space in relativity, uses in computing, etc.) then this format is very appealing. However, it lacks an overarching sense of story that will keep you reading all the way through like the best popular science books can.
What is good to see is they're not afraid of displaying mathematical functions and formulas and then challenging you to calculations that actually mean something, e.g. why calculate the length of a line of a sphere when you can calculate the distance between New York and Sydney on the Earth?
Summary
£400 is a steep price for a comprehensive set of popular maths books, although calling them "popular" perhaps does not do them credit. Think of them more as a set of encyclopaedias of the mathematical world, but with each chapter being eminently more readable than a fact driven summary of events.
There is no better set that will give you the range and depth of subjects available. Due to their diverse nature you will discover topics that interest, entertain and even surprise you. Thus, if you have want to see what mathematics can really do outside of the classroom then these books are you.
[Rant 1] Of course you're going to find things in the golden ratio if you look for it. Any ratio can be found in nature if you look hard enough. I freely admit that in certain cases you can attribute inspiration to the golden ratio, e.g. Le Corbusier's architecture or Dali's and Da Vinci's paintings. However, these are explicit uses. Stating that certain natural phenomena occur in the golden ratio is much dodgier.
To really believe that the ratio is there I would want to understand WHY it is there. This is the essence of mathematical biology. Biologists observe phenomena, mathematicians try to understand its cause. For instance, there are reasons why the golden ratio/ Fibonacci's sequence should appear in sunflower seed packing as it is the optimal packing ratio. Compare this to the claim that your fingers are in the golden ratio and the suggestion that beautiful people are more in the golden ratio than not. Not only are you left asking why but you're suggesting that people who don't fit your rigid proportions are ugly!
The golden ratio supposedly appearing in the human body.
